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<img src="../_static/app_title_images/qmmm-pe.png" alt="QM/MM partitioning in the PE model." class="notebook-image">
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<div class="notebook-entry" data-tags="gpu, dynamics, noiseless">
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<a href="../applications/python/entanglement_acc_hamiltonian_simulation" class="notebook-title">Entanglement Accelerates Quantum Simulation</a>
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<div class="notebook-content">
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Learn how entanglement growth can *reduce* the Trotter error of the first-order product formula (PF1), recovering the result from the [paper](https://www.nature.com/articles/s41567-025-02945-2) using NVIDIA CUDA-Q.
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<img src="../_static/app_title_images/entanglement_acc_hamiltonian_simulation_preview.png" alt="PF1 error decreases as entanglement spreads" class="notebook-image">
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<li class="toctree-l2"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html">Entanglement Accelerates Quantum Simulation</a><ul>
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<li class="toctree-l3"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#2.-Model-Definition">2. Model Definition</a><ul>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#2.1-Initial-product-state">2.1 Initial product state</a></li>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#2.2-QIMF-Hamiltonian">2.2 QIMF Hamiltonian</a></li>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#2.3-First-Order-Trotter-Formula-(PF1)">2.3 First-Order Trotter Formula (PF1)</a></li>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#2.4-PF1-step-for-the-QIMF-partition">2.4 PF1 step for the QIMF partition</a></li>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#2.5-Hamiltonian-helpers">2.5 Hamiltonian helpers</a></li>
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</ul>
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</li>
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<li class="toctree-l3"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#3.-Entanglement-metrics">3. Entanglement metrics</a></li>
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<li class="toctree-l3"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#4.-Simulation-workflow">4. Simulation workflow</a><ul>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#4.1-Single-step-Trotter-error">4.1 Single-step Trotter error</a></li>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#4.2-Dual-trajectory-update">4.2 Dual trajectory update</a></li>
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</ul>
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</li>
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<li class="toctree-l3"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#5.-Reproducing-the-paper’s-Figure-1a">5. Reproducing the paper’s Figure 1a</a><ul>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#5.1-Visualising-the-joint-behaviour">5.1 Visualising the joint behaviour</a></li>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#5.2-Interpreting-the-result">5.2 Interpreting the result</a></li>
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<li class="toctree-l3"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#6.-References-and-further-reading">6. References and further reading</a></li>
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- [What to
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- [Entanglement Accelerates Quantum
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Simulation](../applications/python/entanglement_acc_hamiltonian_simulation.html){.reference
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.internal}
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- [2. Model
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Definition](../applications/python/entanglement_acc_hamiltonian_simulation.html#2.-Model-Definition){.reference
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.internal}
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- [2.1 Initial product
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state](../applications/python/entanglement_acc_hamiltonian_simulation.html#2.1-Initial-product-state){.reference
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- [2.2 QIMF
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Hamiltonian](../applications/python/entanglement_acc_hamiltonian_simulation.html#2.2-QIMF-Hamiltonian){.reference
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- [2.3 First-Order Trotter Formula
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(PF1)](../applications/python/entanglement_acc_hamiltonian_simulation.html#2.3-First-Order-Trotter-Formula-(PF1)){.reference
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- [2.4 PF1 step for the QIMF
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partition](../applications/python/entanglement_acc_hamiltonian_simulation.html#2.4-PF1-step-for-the-QIMF-partition){.reference
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helpers](../applications/python/entanglement_acc_hamiltonian_simulation.html#2.5-Hamiltonian-helpers){.reference
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- [3. Entanglement
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metrics](../applications/python/entanglement_acc_hamiltonian_simulation.html#3.-Entanglement-metrics){.reference
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- [4. Simulation
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workflow](../applications/python/entanglement_acc_hamiltonian_simulation.html#4.-Simulation-workflow){.reference
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- [4.1 Single-step Trotter
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- [5. Reproducing the paper's Figure
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- [5.2 Interpreting the
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- [6. References and further
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- [Backends](../using/backends/backends.html){.reference .internal}
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<li class="toctree-l2"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html">Entanglement Accelerates Quantum Simulation</a><ul>
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<li class="toctree-l3"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#2.-Model-Definition">2. Model Definition</a><ul>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#2.1-Initial-product-state">2.1 Initial product state</a></li>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#2.2-QIMF-Hamiltonian">2.2 QIMF Hamiltonian</a></li>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#2.3-First-Order-Trotter-Formula-(PF1)">2.3 First-Order Trotter Formula (PF1)</a></li>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#2.4-PF1-step-for-the-QIMF-partition">2.4 PF1 step for the QIMF partition</a></li>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#2.5-Hamiltonian-helpers">2.5 Hamiltonian helpers</a></li>
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</ul>
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</li>
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<li class="toctree-l3"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#3.-Entanglement-metrics">3. Entanglement metrics</a></li>
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<li class="toctree-l3"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#4.-Simulation-workflow">4. Simulation workflow</a><ul>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#4.1-Single-step-Trotter-error">4.1 Single-step Trotter error</a></li>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#4.2-Dual-trajectory-update">4.2 Dual trajectory update</a></li>
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<li class="toctree-l3"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#5.-Reproducing-the-paper’s-Figure-1a">5. Reproducing the paper’s Figure 1a</a><ul>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#5.1-Visualising-the-joint-behaviour">5.1 Visualising the joint behaviour</a></li>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#5.2-Interpreting-the-result">5.2 Interpreting the result</a></li>
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<li class="toctree-l3"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#6.-References-and-further-reading">6. References and further reading</a></li>
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- [What to
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- [Entanglement Accelerates Quantum
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Simulation](../applications/python/entanglement_acc_hamiltonian_simulation.html){.reference
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.internal}
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- [2. Model
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- [2.1 Initial product
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state](../applications/python/entanglement_acc_hamiltonian_simulation.html#2.1-Initial-product-state){.reference
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- [2.2 QIMF
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Hamiltonian](../applications/python/entanglement_acc_hamiltonian_simulation.html#2.2-QIMF-Hamiltonian){.reference
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- [2.3 First-Order Trotter Formula
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(PF1)](../applications/python/entanglement_acc_hamiltonian_simulation.html#2.3-First-Order-Trotter-Formula-(PF1)){.reference
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- [2.4 PF1 step for the QIMF
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partition](../applications/python/entanglement_acc_hamiltonian_simulation.html#2.4-PF1-step-for-the-QIMF-partition){.reference
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- [2.5 Hamiltonian
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helpers](../applications/python/entanglement_acc_hamiltonian_simulation.html#2.5-Hamiltonian-helpers){.reference
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- [3. Entanglement
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metrics](../applications/python/entanglement_acc_hamiltonian_simulation.html#3.-Entanglement-metrics){.reference
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- [4. Simulation
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workflow](../applications/python/entanglement_acc_hamiltonian_simulation.html#4.-Simulation-workflow){.reference
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<li class="toctree-l2"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html">Entanglement Accelerates Quantum Simulation</a><ul>
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<li class="toctree-l3"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#2.-Model-Definition">2. Model Definition</a><ul>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#2.1-Initial-product-state">2.1 Initial product state</a></li>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#2.2-QIMF-Hamiltonian">2.2 QIMF Hamiltonian</a></li>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#2.3-First-Order-Trotter-Formula-(PF1)">2.3 First-Order Trotter Formula (PF1)</a></li>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#2.4-PF1-step-for-the-QIMF-partition">2.4 PF1 step for the QIMF partition</a></li>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#2.5-Hamiltonian-helpers">2.5 Hamiltonian helpers</a></li>
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<li class="toctree-l3"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#3.-Entanglement-metrics">3. Entanglement metrics</a></li>
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<li class="toctree-l3"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#4.-Simulation-workflow">4. Simulation workflow</a><ul>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#4.1-Single-step-Trotter-error">4.1 Single-step Trotter error</a></li>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#4.2-Dual-trajectory-update">4.2 Dual trajectory update</a></li>
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<li class="toctree-l3"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#5.-Reproducing-the-paper’s-Figure-1a">5. Reproducing the paper’s Figure 1a</a><ul>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#5.1-Visualising-the-joint-behaviour">5.1 Visualising the joint behaviour</a></li>
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<li class="toctree-l4"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#5.2-Interpreting-the-result">5.2 Interpreting the result</a></li>
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<li class="toctree-l3"><a class="reference internal" href="../applications/python/entanglement_acc_hamiltonian_simulation.html#6.-References-and-further-reading">6. References and further reading</a></li>
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